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.999999999999... = 1 ?


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#121 CrimsonCasio

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Posted 02 October 2004 - 03:04 AM

yeah, we all (at least me) agree now... though i would like to post my teachers example as it was very simple.

#122 R00KIE

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Posted 08 October 2004 - 10:51 AM

You still don't get it.
you (and me) talk of infinity as a number but it is a concept of something larger than anything else, and in case you have forgotten lim x->+inf (1/x)=0+ and not just 0 wich means that it is not really zero but really close, so 0.9(9) != 1.

#123 2072

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Posted 08 October 2004 - 04:13 PM

in 0.999... infinity isn't a number like in lim x->+inf (1/x)

#124 Marco

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Posted 08 October 2004 - 06:36 PM

you (and me) talk of infinity as a number but it is a concept of something larger than anything else, and in case you have forgotten lim x->+inf (1/x)=0+ and not just 0 wich means that it is not really zero but really close, so 0.9(9) != 1.

<{POST_SNAPBACK}>


This doesn?t matter.

Of course, "infinity" is not a number you can calculate with but a concept, as you have an infinitive amount of numbers, but in spite of NO number, that is infinitive great. But this is EXACTLY WHY 0.999... is equal to 1.

"Infinity is not a number" means, it has neither an ancestor ("infinity-1"), nor a successor ("infinity+1"). Thats why numbers like for example "0.000.....1" (infinitive number of digs "0", last digit is "1", however) CAN'T exist: if it would be possible that a digit is at an "infinitive" position, it would not have any value (you said yourself: "infinity is not a number", so a digit can?t be found at position "infinitive", can?t it??). If you would find an infinitive number of digits "0" (what by the way is impossible already, because infinity IS no number), digit "1" would be found at position "infinity+1" than (...er?).

And what would you think then 1-0.999.... is like (as I showed "0.000...1" is impossible)? So no more doubts please B)

By the way, lim x->+inf (1/x)=0+ means NOT, that the result is != 0. It means: "the result would be 0 if x was infinitive, but as x NEVER can be infinitive (because X IS A NUMBER, BUT "INFINITY" IS NOT), the result of 1/x will never be exactly 0, but always a little bit greater than".

---------
PS: I'm going to think now, that numbers like "0.999..." (with "INFINITIVE" digits "9") also can?t exist for even that reason...

0.9 exists, 0.999999 exists, and 0.99999999999999999999999999999999 exists, too, but 0.999... does not. Not only that you can?t draw it or calc with; it DOES NOT exist.

#125 CrimsonCasio

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Posted 08 October 2004 - 07:40 PM

one way to look at it:

1/3= .33333~
2/3= .66666~
3/3= .99999~
but we all know that any number over its-self is 1 so 1=.9999~=3/3

#126 Orwell

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Posted 09 October 2004 - 02:38 PM

I dont agree with Marco.
It's ok that 0.999... = 1, but i cannot say this number doesn't exist (how could it be equal to something that really exists?).
We all agree here to say that 1/3 = 0.333... is a real number, so it does exist, even if it has a infinite number of '3'.

If i follow you, than PI (=3.1415...) with its infinite number of decimals, doesn't exist neither :lol:

Another thing is, that some numbers have or don't have an infinite numbers of decimals, depending on the basis you use.
For exemple, 1/3 = "0.3333..." has infinite decimals if the basis is "decimal", but if you choose 3 as the new basis, you just have to write "0.1".
And btw, "0.45" in decimal mode has no infinite decimals, but in binary mode you will find "0.011100110011001100..." which is a periodic number with infinite decimals ^_^

So the "number of decimals" depends on the basis you use to represent the number itself: you cannot say if it exists or not by seeing that ;)

#127 Marco

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Posted 09 October 2004 - 08:03 PM

If i follow you, than PI (=3.1415...) with its infinite number of decimals, doesn't exist neither


I'd not say that numbers that you would have to represent by an infinitive amount of digits (PI or "0.9999....", for example) do not exist, but their REPRESENTATION (in a decimal system, for example), does not!

Of course, PI exists. But there is NO possible representation as number, no matter what base (2? 10? x?) you use for the digit system, because you would need an infinitive amount of digits always. Represented in digits, "infinitive" long numbers always can be rounded to values only.

Of course you can round PI to a LIMITED amount of decimals, 3, 10, 100, 1000 for example. But that's not PI then, it's a rounded value, only.

Real "PI" is not a number like "3.1415...", but it's the algorithm how to build it. So "3.1415..." with unlimited amount of decimals does not exist, too.

For exemple, 1/3 = "0.3333..."


That's quite similar. The representation of "1/3" as "0.33333..." is not exact, because you either REALLY would have to write down an infinitive number of digits or calculate with an infinitive amount of digits if you want to use it for mathematical operations (what you might see is impossible ;) ), or you have to round it. So the exact description of "1/3" is not a number (not in the DECIMAL system at least; in a system where the base is 3, you could write 0.1 for one third), but the algorithm how to build it: "divide 1 by 3".

And btw, "0.45" in decimal mode has no infinite decimals, but in binary mode you will find "0.011100110011001100..


That`s also the same: you can`t say "0.45 in decimal mode is 0.011100110011... in binary mode", but you have to say "0.45 in decimal mode can`t be represented in binary mode, so you have to round it to 0.011100110011..., but with a limited number of digits".

-----

If you think numbers with an infinitive amount of digits could exist in spite of (no matter what number you use for the base), so simply write one down and send it to me. ;)

#128 huhn_m

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Posted 10 October 2004 - 10:01 AM

so according to your statements you can not calculate with numbers
like "0.3333" and so you can't say

1/3 = 0.333333
3*1/3=3/3=0.999999

Am I right? Although I may now agree with the theorie but because of
other statments made like the lim. thing

#129 CrimsonCasio

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Posted 10 October 2004 - 12:46 PM

that was one of the two examples my math teacher used to show me that ,9=1, there was a better one but for the life of me i cant remeber it.

#130 R00KIE

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Posted 11 October 2004 - 11:53 PM

@huhn_m
now you have made a big mistake

1/3 = 0.333333
3*1/3=3/3=0.999999

1/3=0.33333 i would say that it is half correct but if you said
1/3~=0.33333 then i would agree completely
but 3*1/3=0.9999 is totally wrong because 3*1/3=3/3=1, this one is obvious, no :huh:

#131 huhn_m

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Posted 12 October 2004 - 07:32 AM

you have not read the text above these statements.

I meant of course:

1/3 = 0.33333...
3*1/3=3/3
3*0.33...=0.9999...


and of course we know that 3/3 = 1 but I assumed that you could guess this from the context. Anyways what I wanted to say was, that
accourding to Marcos statments you can not multiply with these (endless) numbers and so this proof is impossible.

#132 Marco

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Posted 12 October 2004 - 08:21 AM

huhn is compleately right:

1/3 = 0.333...
--> 3*1/3 = 3/3 = 1 = 0.999...


is NOT a valid proof, that 1=0.999... is true, because infinite digits do not exist, so you have to round 0.333 and 1/3 <> 0.333...!

What do yo do if you calc "1/3+1/3+1/3"? Do you perform an endless addition? I guess you don't ;)

#133 R00KIE

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Posted 12 October 2004 - 10:17 AM

In that case the calculator is "smart" enough to do the EXACT addition

#134 Marco

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Posted 12 October 2004 - 12:48 PM

In that case the calculator is "smart" enough to do the EXACT addition

<{POST_SNAPBACK}>


It does not! Altough the calculator SEEMS to get the correct result, it even does not calculate with an infinite number of digits, but calculations internal are done some more digits precis than the result that is putted out.

Lets say, it calculates with 5 digits, and puts out a result with 4 digits only. Then "1/3+1/3+1/3" would look like follows:

Enter "1/3"
--> internal stored as "0.3333"
Enter "1/3+1/3+1/3"
--> internal operation: 0.3333+0.3333+0.3333 = 0.9999

Put out 0.9999 with 4 digits precis (last digit of 0.9999 is >= 5, so round upwards):
--> 1.000 on display

#135 Orwell

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Posted 12 October 2004 - 02:38 PM

Ok then, for you, 10 exists but not 10.0000... ?! :huh:

#136 R00KIE

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Posted 12 October 2004 - 04:25 PM

I assumed he was using the fraction mark and not the division sign, otherwise you're right, sorry.

#137 Marco

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Posted 12 October 2004 - 04:53 PM

Ok then, for you, 10 exists but not 10.0000... ?!  :huh:

<{POST_SNAPBACK}>


:blink: What?s this for a question?? :blink:

I think you don?t really understand what I mean...

"10" does not differ from "10.0000" in any way. This are just two ways to represent one and the same number: 10. In "10.0000" you have some digits more, true, but as all these last digits after the point are 0s, they have no value - you never use them. They do not matter for mathematical operations, because they are no real "information". So it?s unimportant how much of them you find.

Thats another thing as in "0.999.." for example, where each digit represents an information that you may NOT simply ignore! Because this would mean, that "0.999" = "0.99", or "41.234 = 41.23", for example, what is definitely not the case; BUT: "10.000" = "10.00" is true, however.

What I simply mean is: an infinitive big information can not exist, because you were unable to process it then. An information that you can?t process has no use for you: how will you work with it? How will you evaluate it? An information that has no use / no content can not be regarded as an information.

And I think you might agree that numbers (and their representation in a decimal or another system) ARE information!

To say this clearly: I NEVER claimed, that "10.0000..." does not exist or that "pi" does not exist, for example. I just claimed, that when the REPRESENTATION of a number like "pi" or "1/3" would require an infitive number of digits in a numbering system (the decimal, for example), this REPRESENTATION does not exist. You ALWAYS have to use a rounded value in such cases in the numbering system you currently use (binary, decimal, ...).

So show me that I?m wrong, but with better arguments than "but my calculator gives the correct result" or "I can write as much 0s as I want after the point".

#138 Overlord

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Posted 12 October 2004 - 05:07 PM

as 1 1.0 1.0000000 1.0000..... 0.9999..... are all representations (aliases) of the same number, the integer "one", there are no problem, because the question was "are the numbers represented by 1 and 0.9999..... equal ?" :)

#139 Orwell

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Posted 12 October 2004 - 06:13 PM

Okay, okay, relax... :D

Don't forget we're on an universal forum, so misunderstanding may happen sometimes (english isn't the native language of everyone, like you certainly know ;) )

So you assume the representation of numbers with infinite decimals don't exist because they contain too much information and "we cannot process it"...

But

an infinitive big information can not exist, because you were unable to process it then. An information that you can?t process has no use for you: how will you work with it? How will you evaluate it? An information that has no use / no content can not be regarded as an information.

What about statistics, probabilities, statistical mechanics etc? What do you think they were made for? But okay, this isn't the subject here ;)

Now why can't i decide myself that the number "'0' + an infine number of decimals '9'" is represented by "0.999..." ? There is no possible confusion, and in fact, when we say "0.999...=1" or "3*0.333...=1", we DO use ALL the information that numbers contain.

There are only conventions we can choose... Everyone understands what i want to say when i write "0.999...". But you're right, thats another thing for "pi", for example...

#140 huhn_m

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Posted 14 October 2004 - 05:28 PM

possible confusion, and in fact, when we say "0.999...=1" or "3*0.333...=1", we DO use ALL the information that numbers contain.


But we do round at a certain point do you agree? There is no possibility to calculate with indefinite numbers and get a non-indefinite number out of i
(except by substracting it from itsself or by multiplying with 0)

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Posted 16 October 2004 - 06:49 PM

0.99999999999...(infinite)=1 :huh:
because <_<
1/9=0.1111111...(infinite) ^_^
0.9999999....(infinite)=9*0.11111111.....(infinite) ;)
9*1/9=1 :)

#142 huhn_m

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Posted 22 October 2004 - 03:34 PM

... you CAN NOT multiply an infinite number since this operation would never
finish. The result is always only an estimation.

Read the post above before posting.

#143 genesis

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Posted 12 February 2005 - 03:43 PM

Well... now that you seem to have agreed on this, may I ask, what are your thoughts on rounding? No one has ever given a compelling arguement to whether a number ending in 5 should be rounded up or down...

I personally round up for two reasons:
1) If 0-4.999... rounds down, then 5-9.999... rounds up (equal amounts in each).
2) If there is a 5, you don't have to look beyond it to see whether you should round up (for example, if your calculator gave the answer 5 but it was 5.0001, it would be wrong to round down.

Then again, you could round to the nearest even number, because if you always round up, the average will go up. But if the numbers were 0.5 and 1.5... And for negative numbers, do you use true value or absolute value?

Also, in science we're supposed to round answers to the number of decimal places/significant figures to the least accurate value in the question but does this change for addition or multiplication? The error is just added in one case, but compounded in the other (our textbooks say addition takes on decimal places and multiplication takes on significant figures...)

#144 liquid

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Posted 12 February 2005 - 11:06 PM

so if .999999999... = 1
and .89999999999999... = .9
ur basically saying any number infinatly close to the next infinatly close number are equal.
Therefore 1=5
and 8=444
being infinatly close divisions between the infinte numbers between any two numbers.......do these two versions of infinity cancel each other out???
please correct me :plol:

#145 Overlord

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Posted 13 February 2005 - 01:46 AM

You can't find "another" number infinitely close to 1 other than ,9999... (1 != 1,000.....0001 because you must stop the number somewhere to write the one, with ,99999... you just continue to write nines without ending) so you can't propage to 1=5

"being infinatly close divisions between the infinte numbers between any two numbers.......do these two versions of infinity cancel each other out???" : sorry, don't understand
in general two infinities doesn't cancel each out : inf - inf != 0 inf/inf != 1
and you can have "strange" things with infinities :

How many elements are there in N (the set of natural numbers : 0; 1; 2; 3; ...) ? Infinity. (a)
How many odd elements are there in N (the set of odd natural numbers : 1; 3; 5; 7; ...) ? Infinity. (B)
How many elements are there in R (the set of real numbers) ? Infinity. ?

But the (a) infinity is "the same" than the (B) infinity (!!!), and ? infinity is "bigger" than (a) infinity or (B) infinity. ^^

#146 liquid

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Posted 13 February 2005 - 06:37 AM

Well actually maby u can have infinate between 1.00000...0001
Think of this paradox tht i cant renember the name for:
If u seperate yourself 10cm from wall and walk 1/2 way there, then 1/2 way again u will never reach it, and if u do this for infinity amount of time, you have passed an infinate distance, even though you are still sumwhere between the wall and the start. You might say that as u approach infinity, your distance covered would be infinatly small, so you will never reach infinate distance, but u are still moving just a bit, so u must go an infinate distance if u did it for infinate time. So basically this means there is infinate distance between any 2 objects. So if you take 1.000 as the start and 000001 as the end you can have an infinate amount of zeros between the two points.

Ok tht sounds stupid and i know its wrong but it does raise questions
If it is wrong then tht must mean that an infinate number of infinatly small divisions between two points must sumhow cancel each other out,,,,,,or that the universe is actually not continous, but discrete at such a small level that we cant tell yet.

Now think about this.....if there is an infinate amount of division between two points then is it possible to advance one of those divisions.....no.....because if you can advance some real distance then there cannot be an infinate amount of divisions

#147 m4x

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Posted 13 February 2005 - 06:31 PM

the universe is actually not continous, but discrete at such a small level that we cant tell yet.

<{POST_SNAPBACK}>


Actually I've heard that scientists presume that not only energy comes in small portions, but also space and time is not continuous, but discrete.

#148 liquid

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Posted 13 February 2005 - 09:33 PM

electron orbitals may be one example af discrete'ness.

#149 m4x

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Posted 14 February 2005 - 09:50 AM

Yes, but there IS space between them. It's just the electrons can't take intermediate positions. What I meant is that some scientists believe there exist a smallest unit of space and time and there is no division possible beyond it. A sort of a basic cell in the grid.
I think we're gettin' a little off topic here :P

#150 liquid

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Posted 14 February 2005 - 11:10 AM

yea i know,,,,i was jst trying to ummmm duh??.......dribble dribble.
0.999999..... = 1 ummmm well if it is infinatle 'close' to 1 then there is also an infinatly small 'gap' between them. For it to equal 1, then an infinatly small gap must equal zero, but if an infinatly samll gap = 0 then the infinate number of infinatly small gaps between any two numbers would = 0 as inf*0 = 0. Then saying sumthing like 4=10 and 6=99 would be true. Thus 0.9999999.... cannot = 1

Another thing to think about......time could not have been around forever as if it did, then it would be impossible to be when we are now, as it would have taken infinity to get this far. Thus time was created. If time was created then how could of 'gods' create the unuverse as there was no time for them to do it. :roflol: ...must be out of the human capability of thinking.

#151 2072

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Posted 14 February 2005 - 06:59 PM

I'm sorry but your logic is flawed...

Read Overlord's post again.

#152 TacoFred

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Posted 14 February 2005 - 08:46 PM

inf*0 is not 0, but undefined
because inf is not some calculatable number
but only a concept of something that never ends

#153 octobclrnts

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Posted 15 February 2005 - 02:26 AM

Looking at the posts in this forum, it occurs to me...WE ARE DORKS. The two things that really seem to push our buttons are politics and number theory. The best part about it is that, I love it and I think most everyone else does too. :) ;)

#154 Simprobe

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Posted 15 February 2005 - 02:33 AM

.50000000000000000000000000000000000000000000000000001 = .5?

LOL..

ROUND 1 .. FIGHT!

#155 liquid

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Posted 15 February 2005 - 05:57 AM

5=5.00000...0001

Now there is always a number infinatly close to the next, so what would be infinatly close to 5 apart from 4.9999999..., this must be 5.0000....00001, otherwise u cannot have an infinatly close number to 5 on the positive side and this would produce a number system that it not continous.

Also u can take the real world as a example. Label 2 points a few cm apart point 5 and 6. u can 'measure'(u cnt really) the distance between these 2 points to an infinate amount af accuracy, but there is still a start '5' and an end point '6'.
Thus 5.0000....0001 , 5 is start point, 0001 is end point, with infinate 000000 in between represents the infinate amout of accuracy.

from this then u can say 5=7 and 6=999 which is plain silly
thus 4.999999... cant equal 5. It is infinatly close to it, but not equal.

#156 Marco

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Posted 15 February 2005 - 08:49 AM

*lol* it's a bit strange but I noticed that quantum physics as well as scientists saying there is a smallest unit of space and time (never heard scientists saying this though; who was it?) seem to support my theory that information cannot be infinitive complex. Because:

a ) You can regard universe to be nothing but information and all physical values occurring in it, too (such as a position in space or time)

b ) But this would mean a position in space or time for example could not be described infinitive exactly, else the information would be "infinitive complex" (and I believe that this is impossible). This either would:

1) Imply a smallest unit of space and time = a scaled, not a continuous one (and there wouldn't be continuous positions, but only multiples of that unit) or

2) Imply an uncertainty, then you also could not describe things any exact you want (and that's exactly what Heisenberg's principle of uncertainty says) or

3) Also 1) and 2) together? Be that as it may.

------
However, I think I know what liquid means: let's call the "infinitive small gap" 1-0.999? = epsilon.

Be epsilon != 0 would mean it's not infinitive small, and there was a number between 0.999? and 1 (because you even could imagine a number between 0 and epsilon). That can't be as epsilon is the smallest gap.

Be epsilon = 0 would mean 1 = 0.999?9. This though would imply 0.999?9 = 0.999?8 = 0.999?7 = ? .So far so good (there's no error in). Liquid means now that continuing this order implies that: 1 = 0.999?9 = 0.999?8 = ... = 0.572333?5 = 0.572333?4 = ? and so on, thus all numbers are the same.

But liquid your error in logic is here: you forget that the digits you change and the highest - value digit there is an infinitive amount of digits between. So changing the last digits has no affect. 1, 0.999?9, 0.999?8 and 0.999?12345 really are all the same. But continuing this order, you never will come to 0.999?12345 = ? = 0.572333...4 or something.

Thus, all numbers are NOT the same. (But anyway, I believe that numbers like 0.999?12345 do not exist :P

---Edit:---

that was a quite bad example; of course 0.999?12345 exists as it's the same as 1 and not an "infinitiv complex" information therefore. But imagine a number with an infinitiv amount of digits following no special construction instruction. 123.328471736582336493989274...0177283492 would be such a number fore example. Note: but numbers like pi, ln(2) or (sin?(3*ln 2)/pi)^e would not, as they follow a special construction instruction).

#157 genesis

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Posted 15 February 2005 - 09:58 AM

b ) But this would mean a position in space or time for example could not be described infinitive exactly, else the information would be "infinitive complex" (and I believe that this is impossible).

I think this is more of a philosophical question than logic.

Logically, infinity*some number=infinity, but a small unit of matter, say an atom, takes up infinitely much space (brings back memories of the 'infinite plane' <_< ). We are composed of atoms, so we take up infinity*some number of space. Is that to say we take up the same amount of space as an atom?

One could also argue there are infinitely many particles in the universe (has it been scientifically proven that there are bounds to the universe?) Also, do waves (eg. radiation) take up space, or only particles? It has not been proven that waves occupy space, so maybe they are the infinite divisions... And our thoughts and ideas, do they take up space? Are they just particles (suggesting that we don't have free-will)?

#158 liquid

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Posted 15 February 2005 - 10:38 AM

But liquid your error in logic is here: you forget that the digits you change and the highest - value digit there is an infinitive amount of digits between. So changing the last digits has no affect. 1, 0.999?9, 0.999?8 and 0.999?12345 really are all the same. But continuing this order, you never will come to 0.999?12345 = ? = 0.572333...4 or something.


Ur right Marco.
But i do think that 1, 0.999?9, 0.999?8 and 0.999?12345 are not the same, even if there if inf numbers in between the start/end. because if u say that, then tht supports the argument that 1=0.9999... in a way.
i think the error in my logic is that even thougth those numbers were infinatly close, that means there would be inf amount of numbers between any two numbers also, so its a two way thing really, i think inf close, and inf numbers in between two numbers sortof cancel each other out.

Well guys how about this lame mans argument:
"=" means equals

0.999999... = 1
one side is compossed of 0.999999..., the other 1, thus they are not equal.

u might say 2x=5 where x=2.5,,,,,yes the two sides are different, but thats algeba whereas 0.99999... and 1 r jst numbers wif no fancy strings attached.

#159 m4x

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Posted 15 February 2005 - 11:21 PM

Well, how about doing some algebra:
0.999999... = 1 / -0.09999... (both sides)
you get:
0.999999... -0.09999... = 1 -0.09999...
0.9 = 0.9 TRUE :D

Well guys how about this lame mans argument:
"=" means equals

0.999999...  =  1
one side is compossed of 0.999999..., the other 1, thus they are not equal.

<{POST_SNAPBACK}>


I don't know if this argument is solid enough. Someone could say that 9/9=1 is false because the sides of the equation differ. You can say that an equation is false only when you lack any way to transform any of it's sides and it is clearly visible that they are unequal.

#160 liquid

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Posted 16 February 2005 - 01:27 AM

ive gotta say i dont really care n'more, this is getting boring.
Anyway u cant do maths with infinate numbers
like 1/3 = 0.3333...
does your calculator figure out all the 33333, no it cant so when u try to prove 0.99999... = 1 and ur using algebra 2 do so, its not right as u are not actually going through and dealing with every 9999 in the number as its impossibe, so u still cant prove 0.99999... =1




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